nLab Archimedean ordered integral domain

Idea

An Archimedean ordered integral domain is an ordered integral domain that satisfies the Archimedean property.

Examples

Archimedean ordered integral domains include

• integers
• rational numbers
• decimal rational numbers
• dyadic rational numbers
• real numbers

Non-Archimedean ordered integral domains include

• p-adic integers.

Properties

Theorem

The integers are the initial Archimedean ordered integral domain.

Theorem

The Dedekind real numbers are the terminal Archimedean ordered integral domain.

For an ordered integral domain, we say that $x$ is apart from $y$ if and only if $x \lt y$ or $x \gt y$.

Theorem

Every Archimedean ordered integral domain $R$ with an element $x$ in $R$ apart from every integer is a dense linear order.

Proof

Since $x$ is apart from every integer, there exists an integer $a$ such that $0 \lt x - a \lt 1$, and since $R$ is Archimedean, it does not have either infinite or infinitesimal elements, which means there exists a natural number $b$ such that $1 \lt b(x - a)$. In addition, $0 \lt x - a \lt 1$ implies $0 \lt (d - c)(x - a) \lt d - c$ and $c \lt (d - c)(x - a) + c \lt d$ for all $c$ and $d$ in $R$. Let $y = (d - c)(x - a) + c$. Since there exists an element $y$ such that $c \lt y \lt d$ for all $c$ and $d$, $R$ is a dense linear order.

Theorem

The Dedekind completion of ordered integral domain $R$ with an element $x$ in $R$ apart from every integer is the integral domain of Dedekind real numbers.

Proof

The Dedekind completion of every dense linear order is the Dedekind real numbers. Since by the previous theorem, $R$ is a dense linear order, the Dedekind completion of $R$ is the Dedekind real numbers.

Theorem

That the Dedekind completion of every Archimedean ordered integral domain is isomorphic to either the integers $\mathbb{Z}$ or the Dedekind real numbers $\mathbb{R}$ is equivalent to the analytic $\mathrm{LPO}$ for $\mathbb{R}$.

Proof

Let $\mathbb{Z}[x]$ denote the free commutative ring on a singleton and let $I$ be a ideal of $\mathbb{Z}[x]$. Consider an Archimedean ordered integral domain $R$ which is ring isomorphic to a quotient ring $\mathbb{Z}[x] / I$, $R \cong \mathbb{Z}[x] / I$, with element $x \in R$. The Dedekind completion of $R$ is the integers if and only if $x$ is equal to an integer, and the Dedekind completion of $R$ is the Dedekind real numbers if and only if $x$ is apart from an integer. However, $x$ is equal to an integer or apart from an integer for all $x \in R$ if and only if $R$ is a discrete integral domain, and the claim that every Archimedean ordered integral domain is discrete is equivalent to the analytic $\mathrm{LPO}$ for the Dedekind real numbers.

Related concepts

• Archimedean ordered field

• Archimedean group

• archimedean difference protoring

• enriched set